John F. Kennedy School of Government

Harvard University Faculty Research Working Papers Series

The views expressed in the KSG Faculty Research Working Paper Series are those of the author(s) and do not necessarily reflect those of the John F. Kennedy School of Government or Harvard University. All works posted here are owned and copyrighted by the author(s). Papers may be downloaded for personal use only.

Strategic Trade and Delegated Competition

Nolan H. Miller and Amit Pazgal

October 2002

RWP02-042

Strategic Trade and Delegated Competition∗

Nolan H. Miller†and Amit Pazgal‡

October 15, 2002

Abstract

Strategic trade theory has been criticized on the grounds that its predictions are overly

sensitive to modeling assumptions. For example, Eaton and Grossman (1986) show that Bran-

der and Spencer’s (1985) seminal result — i.e., when firms compete by setting quantities the

optimal policy involves governments subsidizing their domestic industries — is reversed if the

firms compete by setting prices. Applying recent results in duopoly theory, this paper con-

siders three-stage games in which governments choose subsidies, firms’ owners choose incentive

schemes for their managers, and then the managers compete in the product market. We show

that if firms’ owners have sufficient control over their managers’ behavior, then the optimal

strategic trade policy does not depend on whether firms compete by setting prices or quantities.

In a linear-demand model in which managers are compensated based on a linear combination of

their own firm’s profit plus a (possibly negative) multiple of the rival firm’s profit, the optimal

policy is to subsidize if goods are substitutes and tax if the goods are complements.

∗We thank Chris Avery, Robert Jensen, Asim Khwaja, Dani Rodrik, and Richard Zeckhauser for helpful comments.†John F. Kennedy School of Government, Harvard University. Nolan_Miller@harvard.edu.‡Olin School of Business, Washington University, St. Louis. pazgal@olin.wustl.edu.

1 Introduction

One of the chief shortcomings of strategic trade theory has been that its predictions are highly

sensitive to the modeler’s choice of how to characterize the product market competition. For

example, the main implication of the Brander and Spencer (1985) model — i.e., governments should

subsidize substitute products when firms compete by setting quantities — is reversed if the firms

compete by setting prices (Eaton and Grossman, 1986).1 This disparity leads James Brander to

remark in his 1995 survey of the strategic trade literature that:

The Bertrand [price-setting] model is not necessarily any less plausible than the Cournot

model as an approximation to actual conduct. Because it is hard to know in practice

which of the two models (if either) is appropriate in a given case, the Eaton-Grossman

analysis implies that even finding the sign or direction of the optimal policy might be

difficult. (Brander 1995, p. 1417)

Paul Krugman echoes this sentiment, saying that the “flurry of excitement” over Brander and

Spencer’s original theories had died down:

After several years of theoretical and empirical investigation, it has become clear that

the strategic trade argument, while ingenious, is probably of minor real importance.

Theoretical work has shown that the appropriate strategic trade policy is highly sensitive

to details of market structure that governments are unlikely to get right. (Krugman

1993, p.363)

The dependence of optimal trade policy on the nature of product market competition follows

from the fact that, despite appearing similar, the strategic interaction between two firms who

compete by setting quantities is fundamentally different than the interaction between two firms

who compete by setting prices. A number of papers have studied this distinction (Singh and

Vives (1984); Cheng (1985); Klemperer and Meyer (1986)). They show that when firms produce

differentiated products, the disparity between the price-competition (i.e., Bertrand) and quantity-

setting (i.e., Cournot) outcomes arises from the fact that the elasticity of residual demand facing

a firm whose opponent holds price constant is greater than the elasticity of demand facing a firm1Balboa, Daughety, and Reinganum (2001) provide a recent study of the interaction between assumptions about

market structure and optimal strategic trade policy.

1

whose opponent holds quantity constant, and that price competition is generally “more aggressive”

than quantity competition in the sense that it leads to smaller prices and larger quantities.

The industrial organization literature on strategic delegation explores the idea that altering a

firm’s behavior can alter equilibrium outcomes. Typical models, such as Fershtman and Judd

(1987), Sklivas (1987), Vickers (1985), Fumas (1992), and Miller and Pazgal (2002), consider two-

stage duopoly games in which, during the first stage, profit-maximizing owners choose the incentive

schemes they will give to their managers. During the second stage the managers choose strategic

variables in order to maximize their utility under their incentive schemes. In each case, the owners

have an interest in using the incentive scheme they set to influence their manager’s behavior, which

in turn alters the equilibrium outcome (prices, quantities, and profits) of the two-stage game.

In recent work, Miller and Pazgal (2001), henceforth MP, provide a bridge between the price

vs. quantity competition literature and the strategic delegation literature in which it is shown

that if owners have sufficient control over their managers incentives, then the set of equilibrium

outcomes of a two-stage delegation game does not depend on whether the managers ultimately

compete in prices or in quantities.2 Although players in undelegated price competition behave

more aggressively than players in undelegated quantity competition, when owners can manipulate

their managers’ incentives they make price-setting managers less aggressive and quantity-setting

managers more aggressive, mitigating the difference in behavior, and, if owners have sufficient

control over their managers’ incentives, eliminating it. For example, when demand is linear and

marginal cost is constant, if owners compensate managers based on a linear combination of own

and rival profit, which MP term a relative-performance incentive scheme, the equivalence result

follows.

In this paper we apply the delegated-competition methodology to the strategic trade problem.

In a three-stage game in which governments set subsidies, owners set incentive schemes, and then

managers compete in a third country, we show that once the role of delegation in the owner-

manager relationship is taken into account, the optimal trade policy depends only on the degree of

substitutability of the products, and not on the particular model of product market competition.

When demand is linear, marginal cost is constant, and owners choose relative-performance incen-

tive schemes for their mangers, we show that if products are substitutes, the equilibrium involves

subsidies, while when products are complements, it involves taxes.

2Throughout the paper, we refer to this as the MP equivalence result.

2

In more general contexts, it is computationally difficult to explicitly derive the optimal strategic

trade policy. Nevertheless, it remains the case that the optimal policy does not depend on the

mode of product market competition, provided that owners can exercise sufficient control over their

managers incentives. In light of this, we argue that the sensitivity of the optimal policy instrument

that Eaton and Grossman (1986) identify derives not from the mode of competition, i.e., whether

firms compete in prices or quantities, but rather that these alternative models of competition imply

different behavior on the part of managers. This suggests that conjectural variations models and

other approaches to studying the strategic trade problem that take behavior as primitive may

be more appropriate and ultimately more successful than those that attempt to determine the

correct model of product market competition. Indeed, since there are generally multiple models of

competition that agree with any particular behavior, there may be no such thing as a single correct

model.

The rest of this paper proceeds as follows. In section 2 the optimal trade policy is derived in

the context of a simple example of relative-performance delegation. While still in the context of

the linear model, sections 3 and 4 generalize the result of the example, allowing the goods to be

either substitutes or complements, and discuss the geometric intuition of the problem. Section 5

discusses strategic trade policy in more general contexts. Section 6 concludes. Supporting proofs

are contained in the Appendix.

2 Strategic Trade and Delegation Games

2.1 An Example

We begin with a simple example of the strategic trade game in which the owners of firms may

influence their managers’ behavior. Consider two nations. In each nation, a firm produces a

product, and the two nations’ products are imperfect substitutes. Within each firm, there is an

owner and a manager. The owner is residual claimant on the firm’s net profit, while the manager

makes the actual product market decisions.3 We assume that the products are sold in a third

country. The benefit of this strategy is that we can ignore domestic consumer surplus in our

3We use net profit to refer to the firm’s objective function, profit, net of the subsidy the government imposes. We

use surplus to refer to the government’s objective, which, in the case of third country competition, is the same as

gross profit before the subsidy or tax.

3

measure of national welfare.

We consider three-stage games. In the first stage, the government chooses a per-unit subsidy

(or tax) to be imposed on its domestic firm. In the second stage, the owner of the domestic firm,

knowing the subsidies chosen by both nations, chooses a relative-performance incentive scheme for

its manager. In the third stage, the managers of the firms, knowing the subsidies and incentive

schemes chosen by both sides, choose a value of their strategic variable, i.e., a quantity if the

product market competition is à la Cournot, or a price if it is à la Bertrand.

Since this is a dynamic game of complete information, our equilibrium concept is subgame

perfect Nash equilibrium. We will often refer to the second and third stages of the game as the

“delegation game,” and the equilibrium of the second and third stages, taking subsidies as fixed,

as the “delegation game equilibrium.”

Denote the two nations by 1 and 2. We refer to the government, owner, and manager in nation

i as Gi, Oi, and Mi, respectively. For the purposes of this example, let inverse demand for nation

i’s output be:

pi = 10− qi − 12qj , (1)

where pi denotes good i’s price, qi denotes its quantity, and subscript j = 3 − i denotes thecorresponding variable for other side. The negative coefficient on qj indicates that the products

are substitutes.

Let ci denote the constant marginal cost of production for side i, and note that ci will eventually

consist of the firm’s marginal production cost net of the subsidy set by its government. Hence

ci = c−si, where c is the common marginal production cost of the two firms and si is the side-specificsubsidy chosen by Gi. There are no fixed costs of production.

We assume that the incentive scheme set for Mi takes the form:

mi = πi + viπj ,

where vi is the incentive parameter chosen by Oi, and πi = (pi − ci) qi is the net profit earned byfirm i. We refer to such schemes as relative-performance incentive schemes.

We begin by considering the game under the assumption that product market competition takes

place in quantities. Manager Mi chooses qi in order to maximize:

mi =

µ10− ci − qi − 1

2qj

¶qi + vi

µ10− cj − qj − 1

2qi

¶qj . (2)

4

The optimality condition for the manager’s problem is given by:

−2qi + 10− ci − 12qj − 1

2viqj = 0, (3)

and the equilibrium quantities are found by solving conditions (3) for q1 and q2, yielding:

qi = 230− 10vi − 4ci + (1 + vi) cj

15− vi − vj − vjvi . (4)

Expression (4) gives the third-stage equilibrium quantities as functions of the owners’ incentive

parameters and (implicitly) the governments’ subsidy choices. Evaluating firm i’s net profit at

quantities (4), profit as a function of incentive parameters vi and vj is given by:4

πi = 2 (60 + 10vi − 10vjvi − 8ci + civi + civjvi + 2cj − 2vicj) 30− 10vi − 4ci + (1 + vi) cj(15− vi − vj − vivj)2

. (5)

Next, we move to owner Oi’s optimal choice of incentive parameter vi in the second stage. The

equilibrium values of v1 and v2 (as functions of c1 and c2) are found by solving the following system

for v1 and v2:

d

dv1π1 = 0, and

d

dv2π2 = 0,

where πi is as in (5). This yields optimal values:

v∗i =2ci − cj − 102ci − 7cj + 50 . (6)

Substituting in the optimal values of v1 and v2 from (6) into (4) yields quantities:

q∗i =25

6+1

6cj − 7

12ci, (7)

and profit:

π∗i =1

96(30− 4ci + cj) (50 + 2cj − 7ci) . (8)

Expressions (6), (7), and (8) characterize the equilibrium of the delegation game when managers

compete by setting quantities.

Next, we derive the equilibrium of the delegation game under the assumption that managers

compete by setting prices in the product market. Inverting (1) yields direct demand functions:

qi =20

3− 43pi +

2

3pj . (9)

4Although omitted for the sake of brevity, the second-order conditions for a maximum hold as well.

5

Let Mi be compensated according to the relative-performance incentive scheme, mi = πi + ziπi,

where zi is the incentive parameter in the price-setting version of the delegation game. Manager

Mi chooses pi in order to maximize:

mi = (pi − ci)µ20

3− 43pi +

2

3pj

¶+ zi (pj − cj)

µ20

3− 43pj +

2

3pi

¶.

Differentiating with respect to pi, the optimality condition for Mi’s problem is given by:

pi =5

2− 14zicj +

1

2ci +

1

4(1 + zi) pj . (10)

The third-stage equilibrium prices are found by solving equations (10) for p1 and p2, yielding:

pi =−50− 10zi + 2zicj − 8ci + zjci + cizjzi − 2cj

−15 + zi + zj + zjzi . (11)

Substituting (11) into firm i’s profit, profit as a function of zi and zj is written:

πi =2

3(50 + 10zi − 7ci + cizi − 2zicj + 2cj) ∗ (12)Ã3zicj − zjzicj + 4cj − 30zi − 10zjzi + 2cizjzi − 14ci + 100

(−15 + zi + zj + zizj)2!.

The equilibrium values of the incentive parameters solve:

∂

∂z1π1 = 0, and

∂

∂z2π2 = 0,

where πi is given by (12). Solving this system yields:

z∗∗i =ci − 10

−ci − 30 + 4cj ,

implying equilibrium prices

p∗∗i =1

8cj +

15

4+1

2ci. (13)

Substituting prices (13) into demand equations (9) yields equilibrium quantities and prices:

q∗∗i =25

6+1

6cj − 7

12ci, and (14)

π∗∗i =1

96(30− 4ci + cj) (50 + 2cj − 7ci) . (15)

The equilibrium outcome of the delegation game when firms compete by setting quantities,

given by q∗i and π∗i in (7) and (8), is the same as the equilibrium outcome of the delegation game

6

when firms compete by setting prices, given by q∗∗i and π∗∗i in (14) and (15). Thus the outcome of

the delegation game does not depend on the form of product market competition. Further, notice

that the outcomes are identical as functions of the firms’ costs, and, by extension, as functions of

the governments’ choices of subsidies. And, since the outcome of the delegation game completely

determines the value of the government’s objective function in the first stage, it must be that the

optimal subsidy policy is the same regardless of the form of product market competition. This is

the essence of our results, which we develop in more generality throughout the remainder of the

paper.

We now turn to the first stage of the game, in which the government chooses subsidy rates

in order to maximize home-country welfare given that owners and managers will play equilibrium

strategies in the delegation game. For the rest of this example, let c = 2. Hence ci = (2− si).Making this substitution, the equilibrium quantities and profits in the delegation game (as captured

by (7) and (8) or (14) and (15)) can be expressed as functions of the subsidies only as:

qi =

µ10

3− 16sj +

7

12si

¶, and

πi =1

96(40− 2sj + 7si) (24 + 4si − sj) .

Domestic welfare is given by home-country net industry profit less subsidies paid: wi = πi − siqi.The optimal subsidy rates are found by solving the following system of optimality conditions:

d

dsi

µ1

96(40− 2sj + 7si) (24 + 4si − sj)− si

µ10

3− 16sj +

7

12si

¶¶= 0.

Solving this system yields s∗1 = s∗2 =855 for the parameter values in this example.

5 Thus, in

equilibrium, each country subsidizes its home industry. This is true regardless of whether product

market competition takes place in prices or quantities.

3 Substitutability and Optimal Policy

We now consider a more general model that allows for the goods to differ in their degree of sub-

stitutability. However, for computational efficiency we maintain the assumptions that demand is

linear and symmetric with constant (and equal) marginal cost. The three-stage game remains the

same as in Section 2. Consequently, we suppress many of the calculations. Complete derivations

are available from the authors upon request.5Again, the second order conditions are easily verified.

7

In order to allow for different levels of substitutability (or even complementarity), we consider

inverse demand system:

pi = a− qi + gqj ,

where we assume a > 0 and |g| < 1 so that both price and quantity competition have well-definedsolutions. Marginal cost for each firm is given by ci = c − si.6 As a regularity condition, we

assume a − c > 0, which assures that if firm j is not active, i.e., qj = 0, then firm i wishes to

produce a positive quantity.

Solving the three-stage game as in the example, the equilibrium trade policy is described in

Proposition 1.

Proposition 1 If the goods are substitutes (−1 < g < 0) then the equilibrium trade policy involves

subsidization of the domestic industry. If the goods are unrelated, free trade is optimal. If the

goods are complements (0 < g < 1) then the equilibrium subsidy policy involves taxing the domestic

industry.

Proof. Following the same procedure as above, the equilibrium subsidies in the three stage

game are given by:

s∗i = − (a− c)g3

8− 4g2 + g3 .

For |g| < 1, the denominator is positive, from which it follows that the sign of the optimal subsidy

is opposite that of g.

The main difference between Proposition 1 and previous results in strategic trade theory such

as Brander and Spencer (1985) and Eaton and Grossman (1986) is that the optimal policy depends

only on whether the goods are substitutes or complements and not on the form of product market

competition.

Figure 1 plots s∗ia−c as a function of g, where the normalization by (a− c) controls for market

size. The magnitude of the optimal subsidy increases as the goods become closer substitutes,

reaching 13 (a− c) when the goods are perfect substitutes. When the goods are unrelated, g = 0,

free trade is optimal. As g becomes positive, it is optimal to tax the domestic industry, with the

optimal tax reaching 15 (a− c) as g approaches 1.

6The delegation-game equivalence result also holds for the general linear demand system pi = ai− biqi + giqj andfirm-specific marginal cost ci. The restriction here to symmetric demand and marginal cost is merely for convenience.

8

-0.2

-0.1

0

0.1

0.2

0.3

normalized subsidy

-1 -0.5 0.5 1g

Figure 1: Magnitude of the optimal subsidy and the degree of substitutability

Although we omit the analysis for the sake of brevity, the equilibrium of the three-stage game

resembles the Brander and Spencer (1985) model of strategic trade with substitutes in a number

of ways beyond the optimality of subsidies. For example, as in Brander and Spencer, the strategic

trade game has the flavor of a Prisoners’ Dilemma: each firm has a dominant strategy to deviate

from free trade, but it would be Pareto superior if both nations could commit to free trade policies.

Further, when taxation is distortionary (i.e., $1 in subsidy costs the government $k >1 to raise),

the curve in Figure 1 shifts downward as k increases. Free trade and/or taxation becomes the

equilibrium policy for an interval g ∈ [g (k) , 1], where g (k) is a strictly decreasing function of k.7

4 Intuition for the Result

4.1 The Standard Strategic Trade Game

To understand Proposition 1, it is instructive to begin by revisiting the intuition behind the optimal

policy in the pure (i.e., without delegation) quantity- and price-competition versions of the problem.

Throughout the section, we confine ourselves to the linear environment of Section 3.

When firms compete in quantities in a free-trade environment, each point on a firm’s reaction

7See, for example, as in Neary (1991; 1994) and Gruenspecht (1988), for earlier approaches to the problem of

distortionary taxation. Complete analysis the claims in this paragraph are available from the authors upon request.

9

R1 R1

*

R2

Q2

Q1

Π1

Figure 2: Optimal policy under quantity competition

function maximizes its profit, given the other firm’s strategy choice. Thus, as in Figure 2, the

isoprofit line for player 1 through any point on its reaction function, R1, is flat. The free-trade

equilibrium is at the point where firm 1’s reaction function, R1, intersects firm 2’s reaction function,

R2.

The basic insight of the Brander and Spencer (1985) analysis is that, since R2 slopes downward

for quantity competition in substitutes, moving down and to the right along R2 increases player 1’s

profit, at least for a time. Choosing a positive subsidy makes firm 1 more aggressive, shifting its

reaction curve right to R∗1 and moving the equilibrium outcome down and to the right along player

2’s reaction curve. Since this increases profit, the government prefers setting a positive subsidy

to free trade (or taxation). Indeed, both firms share this incentive, and it persists even when the

other firm sets a non-zero subsidy. Hence the equilibrium involves each government subsidizing its

domestic industry (see Figure 3).

When firms compete by setting prices, the opposite result obtains. For ease of comparison and a

new perspective on the difference between price and quantity competition, we present the projection

of the price-setting game into the quantity space in Figure 4. The free-trade equilibrium is given

by the intersection of firm 1’s price reaction function, r1, with firm 2’s price reaction function, r2.

In this linear case, it is straightforward to show firm 1’s optimal response in the price-setting game

10

R1

*

R2*

Q2

Q1

Π1

Π2

Figure 3: The equilibrium of the delegation game.

is larger than its optimal response in the quantity-setting game (Singh and Vives, 1984), that r2

is flatter than r1, and that firm 1’s profit isoquant is steeper than r2 at the intersection of r1 and

r2 (i.e., the free trade equilibrium). Consequently, beginning at the intersection of r1 and r2, firm

1’s profit increases as the outcome moves up and to the left along r2.

Based on the geometry of the problem, the G1 can increase domestic surplus by shifting firm

1’s reaction curve inward. The way to do this is to tax the domestic firm, which makes it “less

aggressive” and shifts r1 back toward r∗1. Again, these incentives are qualitatively unchanged if

country 2 chooses a non-zero tax, and therefore the equilibrium involves taxation by each nation.

The difference in the direction of beneficial strategic trade policy in Cournot vs. Bertrand

competition has to do with the difference in the relative slopes of the profit isoquants through the

equilibrium point and the slope of the other firm’s reaction function. In quantity competition, the

profit isoquant is flatter, and subsidies improve welfare. In price competition, on the other hand,

the isoprofit curve is steeper, and taxation in beneficial. This comparison is at the heart of the

Eaton and Grossman (1986) analysis, and will continue to be central to our analysis.

11

r1 r1

*

r2

Q2

Q1

Π1

Figure 4: Optimal policy under price competition

4.2 The Delegated Strategic-Trade Game

We now turn to understanding the role of strategic trade policy in the three-stage game. In the

third stage, managers optimize their payoffs, given the incentive schemes chosen by each firm and

the strategy chosen by the other manager. This defines the managers’ reaction functions. In the

second stage, the owners of the firm choose their incentive parameters in order to maximize profit,

given the incentive parameters chosen by the other owner and the resulting third-stage equilibrium.

Holding fixed the other owner’s incentive parameters, each owner chooses the incentive parameter

that results in the third-stage equilibrium that maximizes its profit. The geometric implication

of this is that, at the equilibrium choice of incentive parameters, firm 1’s isoprofit line through

the equilibrium is tangent to firm 2’s reaction curve, and vice versa. This situation is (partially)

depicted in Figure 5, where DRi indicates firm i’s equilibrium reaction curve in the delegation

game.

When the firms choose incentive parameters optimally, the equilibrium outcome is the point

along firm 2’s reaction curve that maximizes firm 1’s profit (and vice versa). Hence, shifting

firm 1’s reaction curve through strategic trade policy cannot increase domestic surplus unless it

also influences firm 2’s equilibrium incentive parameters (i.e., moves DR2). However, since each

12

DR1

*

DR2*

Q2

Q1

Π1

Figure 5: The equilibrium of the strategic-trade game (no delegation).

owner’s optimal choice of incentive parameters depend on the subsidies chosen by both governments,

varying the subsidy influences both firms’ second-stage reaction functions.

Examining the optimal incentive parameters v1 and v2, given by (6), it is easily shown thatdvidsi

< 0 and dvidsj

> 0.8 Therefore an increase in s1 pivots firm 1’s optimal reaction curve outward

and firm 2’s optimal reaction curve downward, as in Figure 6. Hence, following an increase in

s1, the equilibrium point moves to the right of DR1 and below DR2 to the intersection of DR∗1and DR∗2. Since firm 1’s isoprofit line was originally tangent to DR2, this necessarily increases

profit. Subsidizing the domestic industry improves domestic surplus, and the optimal policy

involves subsidization.

Discussion of the simultaneous determination of equilibrium trade policy in the three-stage

game is slightly more complicated, but has the same basic form. Suppose that G2 chooses its

8Substituting ci = 2 − si into (6) yields the following expressions for the optimal incentive parameters, given asfunctions of the subsidies chosen by both nations:

v1 =8 + 2s1 − s2

−40 + 2s1 − 7s2v2 =

8 + 2s2 − s1−40 + 2s2 − 7s1 .

13

DR1

DR2

Q2

Q1

Π1

DR2*

DR1*

Figure 6: Both reaction curves shift with a change in subsidy.

equilibrium subsidy, s∗2, and G1 chooses free trade, s1 = 0. Given that G1 does not intervene in

the market, O1 chooses v1 in order to maximize its true profit. Consequently country 1’s isoprofit

line is tangent to manager 2’s reaction curve when s1 = 0 and s2 = s∗2. As before, increasing s1

shifts M1’s reaction curve to the right and M2’s reaction curve downward (through the influence of

the subsidy on v2). Due to the initial tangency of country 1’s isosurplus and country 2’s reaction

curve, this increases country 1’s surplus.9

When the goods produced by the firms are complements, the firm’s net profit increases with

the other firm’s quantity, and consequently the firm’s isoprofit curves bend upward for firm 1 (resp.

outward for firm 2) instead of downward (resp. inward), as illustrated in Figure 7. Although this

reversal changes the sign of the optimal policy, the remaining geometric relations do not change.

The equilibrium of the delegation game under free trade is located at the intersection of DR1 and

DR2, with firm 1’s profit isoquant tangent to DR2 at the equilibrium point. When G1 introduces

a positive tax on its home industry, O1 responds by pivoting its manager’s reaction curve inward:

the manager becomes less aggressive. At the same time, O2 responds by making its manager more

aggressive, pivoting its reaction curve upward. The result is that the equilibrium of the delegation

9Note that this argument holds even though firm 1’s reaction curve when s1 = 0 and s2 = s∗2 is not the same as

its reaction curve when s1 = s∗1 and s2 = s∗2.

14

DR1

DR2

Q2

Q1

Π1

DR2*

DR1*

Figure 7: The case of complements.

game moves up and to the right, increasing nation 1’s surplus, and the equilibrium of the delegated

strategic trade game with complements involves taxes by both governments.

5 Strategic Trade Theory and General Delegation Games

The discussion in the previous section has focused on the case of linear demand, constant marginal

cost, and relative-performance incentive schemes. Under these conditions, there is a unique equi-

librium of the delegation game, and the optimal trade policy is unambiguous. In more general

environments, the delegation game may have multiple equilibria, each of which is supported by

complicated incentives. Nevertheless, the MP equivalence result continues to hold in the delega-

tion game: if owners have sufficient control over their managers’ incentives, then for any equilibrium

of the delegation game when managers set prices there is a corresponding equilibrium of the del-

egation game when managers set quantities that results in the same final prices and quantities

(and vice versa). Consequently, it remains true that if owners have sufficient power to control

their managers’ incentives, then the optimal trade policy does not depend on the nature of product

market competition.

We relegate the formal statement and proof of this result to the Appendix. Here, we focus on

15

a less formal discussion of what it means for owners to have sufficient control over their managers’

incentives. In order to compare price and quantity competition, MP define an outcome set as

the projection of a manager’s best response correspondence (i.e., reaction curve) into the four-

dimensional (q1, q2, p1, p2)-space. The key condition, which MP denote Outcome Set Equivalence

(OSE), is that the set of behaviors (i.e., outcome sets) the owner can induce on the part of its

manager must be the same regardless of whether the managers choose prices or quantities.10

If OSE holds, then the difference between price- and quantity-competition in the final stage of

the delegation game amounts to nothing more than a difference in the naming of outcome sets.

The fundamental game is unchanged, and consequently the set of equilibrium outcomes of the

delegation game does not depend on the form of the product market competition.

Proposition 2 If OSE holds, then, for any choice of subsidies by the governments, the set of

equilibrium outcomes of the delegation game is the same regardless of whether the firms compete in

prices, quantities, or one firm chooses price and the other chooses quantity.

Proof. See the Appendix.

Proposition 2 implies that the set of possible equilibria and outcomes of the delegation game

that can arise given particular subsidy choices does not depend on the form of product market

competition. Thus, even when there are multiple equilibria in the delegation game, if OSE holds

then there is no equilibrium outcome that can arise under one type of product market competition

and not the other. And since, just as in the earlier results, the governments care about their

subsidy choices only inasmuch as they affect the equilibrium prices and quantities resulting from

the delegation game, the fact that the outcomes are independent of the form of product market

competition implies that the equilibrium choices of strategic trade policies are independent of the

form of product market competition as well.

10Outcome sets (or some similar construction) are necessary to compare price and quantity competition reaction

curves because price reaction curves lie in (p1, p2)-space, while quantity reaction curves lie in (q1, q2)-space. However,

since fixing any two elements of p1, p2, q1, q2 determines the other two, price and quantity reaction curves arecomparable once they are cast in the right frame of reference. Looking at the four-dimensional outcome set is one

way to do so. Another would be to project price reaction curves into the quantity space, as we do in the geometric

analysis. Thus price reaction curve ri generates the same outcome set as quantity reaction curve Ri if and only if

the projection of ri into (q1, q2)-space coincides with Ri.

16

Proposition 3 If OSE holds, then the equilibrium trade policies do not depend on the form of

product market competition.

Proof. See the Appendix.

Proposition 3 does not provide a method of determining the equilibrium trade policy or even

whether subsidizing its domestic industry helps or harms a country. However, if we take seriously

the possibility of delegation, it suggests that a “correct” model of the product market should not

be necessary in order to answer this question. To put it another way, any conclusion that can be

drawn about trade policy knowing the true mode of product market competition is also consistent

with any other model of (delegated) competition that leads to the same managerial behavior, and

knowing how managers’ behavior changes in response to a subsidy change is sufficient to determine

the direction of beneficial change in subsidy policy, even if the government does not have access to

the true model of product market competition.

Our OSE condition is strong and would be hard to verify in any practical situation. However,

while this is true, it would also be difficult to falsify without the type of detailed knowledge of

the inner workings of firms and product markets that is widely believed to beyond the grasp of

government and academic analysts. That is, suppose it were proposed that managers in a particular

strategic-trade problem competed by setting quantities. Propositions 2 and 3 establish that the

possibility that the observed market behavior arose from delegated price competition could not be

ruled out using only data on market prices and quantities. Thus, in addition to establishing that

determining equilibrium strategic trade policies cannot depend on the model of product market

competition, they also establish that any analysis that claims to establish the “correct” model of

product market competition must do so using data other than market outcomes.

6 Conclusion

In this paper, we have argued that the theoretical case for strategic trade policy is not as flawed

as the statements from Brander and Krugman quoted in the introduction would suggest. If

owners have sufficient control over their managers’ incentives, then the optimal/equilibrium trade

interventions do not depend on the nature of product market competition. In the linear model,

we showed that owners’ ability to set simple, relative-performance incentive schemes is sufficient

for the invariance result to hold. In more complicated environments, more complicated incentive

17

schemes may be required, but the basic result is robust.

The most general insight to be taken from the results derived here is that, at its base, equilibrium

depends on the behavior of managers and not on whether that behavior derives from delegated price-

or quantity-setting. Therefore, even though, as Brander suggests, it may be difficult to determine

the right model of product market competition, this is not really necessary to solve the strategic

trade problem. In fact, our model suggests that, from a descriptive perspective, there may be no

such thing as the “correct model.” For this reason we argue that, in approaching the strategic

trade problem, the primitive notion should be the behavior of the product market, not the model

of the product market.

Our conclusions offer support to Eaton and Grossman’s (1986) conjectural variations-based

model of strategic trade and to the empirical strategic trade literature that relies on conjectural

variations.11 The conjectural variations approach is often criticized on the grounds that players’

conjectures may be inconsistent in that they posit behavior on the part of their opponents that is

not confirmed in the equilibrium. Our delegation approach avoids this criticism. In our model,

managers’ behavior is optimal given the incentive schemes set by owners, and owners’ incentive

schemes are optimal given the trade policies adopted by the governments. Hence, the managers’

conjectures are, in fact, consistent once the owners’ and governments’ strategies are taken into

account. And, since the owners and governments are both optimizing, no inconsistency between

conjectures and behavior remains anywhere in the model.

11See Krugman (1989;1212-1213) and Krugman (1994; 1-9) for discussions of empirical strategic trade papers

employing the conjectural variations approach, and various essays in Krugman (1994) for studies employing the

methodology.

18

References

[1] Balboa, O., Daughety, A. and J. Reinganum (2001) “Market Structure and the Demand for

Free Trade,” Varnderbilt University Department of Economics Working Paper 01-W12.

[2] Brander, J. (1995) “Strategic Trade Policy,” in Handbook of International Economics, Volume

3, G. Grossman and K. Rogoff (eds.). Amsterdam: Elsevier, pp. 1295-1455.

[3] Brander, J. and B. Spencer (1985) “Export Subsidies and International Market Share Rivalry,”

Journal of International Economics 18, pp. 83-100.

[4] Cheng, L. (1985) “Comparing Bertrand and Cournot Equilibria: A Geometric Approach,”

RAND Journal of Economics 16, pp. 146-152.

[5] Eaton, J. and G. Grossman (1986) “Optimal Trade and Industrial Policy Under Oligopoly,”

Quarterly Journal of Economics 101, pp. 383-406.

[6] Fershtman, C., and K. Judd (1987) “Equilibrium Incentives in Oligopoly,” American Economic

Review 77, pp. 927-940.

[7] Fumas, V.S. (1992) “Relative Performance Evaluation of Management: The Effects of Indus-

trial Competition and Risk Sharing,” International Journal of Industrial Organization 10, pp.

473-489

[8] Gruenspecht, H. (1988) “Export Subsidies for Differentiated Products,” Journal of Interna-

tional Economics 24, pp. 331-344.

[9] Klemperer, P. and M. Meyer (1986) “Price Competition vs. Quantity Competition: The Role

of Uncertainty,” RAND Journal of Economics 17, pp. 618-638.

[10] Krugman, P. (1989) “Industrial Organization and International Trade,” in Handbook of Indus-

trial Organization, Volume 2, R. Schmalensee and R. Willig (eds.). Amsterdam: Elsevier, pp.

11979-1223.

[11] Krugman, P. (1993) “The Narrow and Broad Arguments for Free Trade,” American Economic

Review, Papers and Proceedings 83, pp. 362-366.

19

[12] Krugman, P. (1994) “Introduction,” in Empirical Studies of Strategic Trade Policy, P. Krug-

man and A. Smith (eds.). Chicago: The University of Chicago Press.

[13] Miller, N. and A. Pazgal (2001) “The Equivalence of Price and Quantity Competition with

Delegation,” RAND Journal of Economics 32, pp. 284-301.

[14] Miller, N. and A. Pazgal (2002) “Relative Performance as a Strategic Commitment Mecha-

nism,” Managerial and Decision Economics 23, pp. 51-68.

[15] Neary, P. (1991) “Export Subsidies and Price Competition,” in International Trade Policy,

Helpman E. and A. Razin (eds.), Cambridge, MA: MIT Press.

[16] Neary, P. (1994), “Cost asymmetries in international Subsidy Games: Should Governments

Help Winners or Losers?” Journal of International Economics 37, pp. 197-218.

[17] Singh, N. and Vives, X. (1984) “Price and Quantity Competition in a Differentiated Duopoly,”

RAND Journal of Economics 15, pp. 546-554.

[18] Sklivas, S. (1987) “The Strategic Choice of Managerial Incentives,” RAND Journal of Eco-

nomics 18, pp. 452-458.

[19] Vickers, J. (1984) “Delegation and the Theory of the Firm,” Economic Journal (Supplement)

95, pp. 138-147.

20

A Development of Propositions 2 and 3

In this section, we extend the MP equivalence result to the strategic trade context, proving that if

owners have sufficient control over their managers’ incentives, then the equilibrium subsidies do not

depend on the form of product market competition. The derivation consists of two parts. First,

following Proposition 3 in MP, we establish that when owners have sufficient control over their

managers’ incentives, the set of equilibrium outcomes of the delegation game, parameterized by

the governments’ choices of trade policies (s1, s2), does not depend on the form of product market

competition. Second, we show that the equilibrium choices of trade policies do not depend on the

form of product market competition.

We begin by making precise what is required for owners to have “sufficient control” over their

managers’ incentives. Throughout this section, r, t, x, y will stand for elements of the set p, q andwill be used to denote either price or quantity competition in various contexts.12 Let qi (p1, p2) be

the demand function for product i, Θri ⊆ <k (where k is a positive integer) be the set of incentiveparameters available to owner i when her firm competes by setting r, and θri be a generic element

of Θri . Let Ui (p1, p2, q1, q2|θri , tj , s1, s2) be manager i0s utility function conditional on owner i0sincentive parameter choice, θri , strategy choice tj by manager j, and subsidy choices s1 and s2.

Holding fixed the governments’ subsidies, s1 and s2, let the outcome set for manager i consist

of all price-quantity quadruples that:

i) are consistent with the demand system

ii) represent a utility maximizing choice for manager i given the strategic choice of manager j,

the incentive parameters chosen by owner i, and the two governments’ subsidy choices.

The outcome set represents the set of price-quantity quadruples that could occur given that

manager i responds optimally to the incentives he is given. If firm i competes by setting r ∈ p, qand firm j competes by setting t ∈ p, q, denote the outcome set for player i by:

Ωrti (θri |s1, s2) =

(p1,p2, q1, q2) : q1 = q1 (p1, p2) , q2 = q2 (p1, p2) , and

ri ∈ argmaxUi (p1, p2, q1, q2|θri , tj , s1, s2) .

12We will use si to denote the particular strategy choice by manager i when he competes by setting s. For example,

when s = p, si = pi stands for the particular price he chooses.

21

The set of third-stage equilibrium outcomes when incentive parameters θri and θtj are chosen is

given by the intersection of the two managers’ outcome sets Ωrti (θri |s1, s2) ∩ Ωtrj

¡θtj |s1, s2

¢.

Now consider the second-stage equilibrium. Owners choose incentive parameters in order to

maximize profit subject to the constraint that the resulting prices and quantities comprise a third-

stage equilibrium outcome, given the choice of incentive parameters by the other owner and the

governments’ subsidy choices. Assuming owner i sets r ∈ p, q and j sets t ∈ p, q in the secondstage, in the first stage the owner solves:

maxθri∈Θri

(pi − ci) qi (16)

subject to (p1, p2, q1, q2) ∈ Ωrti (θri |s1, s2) ∩ Ωtrj³θtj |s1, s2

´,

where θtj is firm j0s equilibrium incentive parameter choice.

A sufficient condition for the equivalence result to hold is that, holding the rival manager’s

behavior fixed, the set of behaviors (i.e., outcome sets) the owner can induce on the part of its

manager must be the same regardless of whether the managers choose prices or quantities.

Formally, this condition can be stated as:

Outcome Set Equivalence (OSE). For player i ∈ 1, 2, for any s, t, x, y ∈ p, q, and any θsi ∈ Θsi ,there exists a θxi ∈ Θxi such that Ωxyi (θxi ) = Ωsti (θsi ).

If condition OSE holds, the distinction between price, quantity, and mixed competition reduces

to mere differences in the naming of outcome sets. The equivalence of outcomes in the delegation

game follows immediately.

Proposition 2: If OSE holds, then, for any choice of subsidies by the governments, the set of

equilibrium outcomes of the delegation game are the same regardless of whether the firms compete

in prices, quantities, or one firm chooses price and the other chooses quantity.

Proof: Suppose that firm i sets r and firm j sets t in the second stage competition. We will

show that the same prices and quantities are an equilibrium outcome when firm i sets x and firm

j sets y. firm i0s profit maximization problem is:

maxθri∈Θri

(pi − (ci + si)) qi (17)

subject to (p1, p2, q1, q2) ∈ Ωrti (θri ) ∩ Ωtrj³θtj

´.

22

The firm does not directly care about the incentive parameters; only the prices and quantities are

payoff relevant. The set of feasible prices and quantities is given by:n(p1, p2, q1, q2) : (p1, p2, q1, q2) ∈ Ωrti (θri ) ∩Ωtrj

³θtj

´for some θri ∈ Θri

o. (18)

By OSE : i) there exists a θyj ∈ Θyj such that Ωtrj

³θtj

´= Ωyxj

³θyj

´, and ii) the set of feasible

outcome sets for manager i when i sets r and j sets t are identical to the set of feasible outcome

sets for manager i when i sets x and j sets t. Hence (18) is identical to:n(p1, p2, q1, q2) : (p1, p2, q1, q2) ∈ Ωxyi (θxi ) ∩ Ωyxj

³θyj

´for some θxi ∈ Θxi

o.

Since the feasible set in

maxθxi ∈Θxi

(pi − (ci + si)) qi

subject to (p1, p2, q1, q2) ∈ Ωxyi (θxi ) ∩Ωyxj³θyj

´is the same as in (17) and only prices and quantities are payoff-relevant, firm i must choose an

incentive parameter that results in the same prices and quantities as in (17), provided that one

exists. By OSE, there exists a θxi ∈ Θxi such that Ωrti

¡θri

¢= Ωxyi

³θxi

´, which implies the same

prices and quantities, and so θxi is a best response to θ

yj . Reversing the roles of i and j completes

the proof. ¥

While OSE is sufficient for Proposition 2, it is by no means necessary. MP decompose OSE

into two weaker conditions, termed Replication and Feasibility, that also imply Proposition 2. See

MP pp. 292-293 for details.

Proposition 3: If OSE holds, then the equilibrium trade policies do not depend on the form of

product market competition.

Proof: Suppose that M1 competes by setting x ∈ p, q in the product market, while M2competes by setting y ∈ p, q. Let Qxy (s1, s2) denote the (non-empty) set of equilibrium quantity

vectors of the delegation game. In the first stage, the Gi chooses si in order to maximize domestic

welfare, subject to the constraint that the resulting prices and quantities comprise an equilibrium of

23

the delegation game. Hence Gi’s problem is written:

maxsi(pi − ci) qi (19)

subject to (q1, q2) ∈ Qxy (si, s3−i) , and

qi = qi (p1, p2) .

A pair of trade policies (s∗1, s∗2) (along with the resulting equilibrium of the delegation game) comprise

an equilibrium of the three-stage strategic trade game if s∗i solves (19) when s3−1 = s∗3−i, for

i = 1, 2.

By Proposition 2, Q (s1, s2) is invariant to the form of product market competition whenever

Replication and Feasibility hold. Hence Qxy = Qrt for r, t, x, y ∈ p, q, from which Proposition 3

is immediate.

24